Curve sketching and optimization

Find the relative maximum and minimum points of $f(x)=x^3-27x+4$ and the corresponding values.

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Find the intervals of monotonicity and the extrema of $f(x)=x{e}^{-x}$.

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Find the intervals of convexity and concavity and the inflection point of $f(x)=x^3-6x^2+2$.

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Find the inflection points of $f(x)=e^{-x^2}$.

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Classify the stationary points of $f(x)=x^4-4x^3$, using the second derivative test where possible.

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Find the asymptotes and the relative extrema of $f(x)=\frac{x^2+9}{x}$.

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Study the function $f(x)=\frac{\ln x}{x^2}$: asymptotes, extrema and inflections.

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Find the absolute maximum and minimum of $f(x)=x^3-3x^2$ on the interval $[-1,4]$.

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The profit of a firm is $\pi (q)=-q^3+9q^2-15q-10$, and the production capacity allows $0\le q\le 4$. Find the maximum profit and the minimum profit on the admissible interval.

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A rectangular enclosure of area $50{\mathrm{m}}^2$ has one side against a wall and the other three sides made of fencing. Find the dimensions that minimize the length of the fencing.

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Among the cylinders of volume $54\pi \mathrm{c}{\mathrm{m}}^3$ find the one of minimum total surface area.

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A firm in perfect competition sells at the price $p=29$ euros and has cost $C(q)=\frac{q^3}{3}-4q^2+20q+30$. Find the quantity that maximizes profit and the maximum profit.

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The production cost is $C(q)=2q^2+8q+200$, with $q>0$. Find the quantity that minimizes average cost and the minimum average cost.

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The demand for a good is $q(p)=108-p^2$, with $0\le p\le \sqrt{108}$. Find the price that maximizes revenue, the maximum revenue and the elasticity of demand at that price.

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